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User blog:Xanderblack3000/My Singing Mathsters 1
Since I'm interested in both math and MSM, why not do a fusion! MSMath 1: When DoF has every single possible monster, how many monsters will it have? We already know that MSM has 30 monsters, but what about DoF? How many monsters will it have if BBB adds every possible monster that can be made using the six elements in the game (Plant, Cold, Air, Water, Earth, Fire)? The one-element monsters are easy to calculate: There's six of them, because there's 6 elements. The two-element monsters are trickier. You might think that there are 30 monsters, because there are six choices for the first element and five choices for the second (because you can't have a plant+plant monster), and six times five equals thirty, but that's not entirely correct. Instead of thinking about it like that, imagine six islands. You need to make bridges that connect each island to each other island. Let's just call them Plant Island, Cold Island, Air Island, Water Island, Earth Island and Gold Island. You need to make 5 bridges to connect Plant Island to Cold, Air, Water, Earth and Gold islands. Now you need to connect bridges between Cold Island and all other islands, except Plant, because you already connected Plant and Cold islands. So now we have 4 more bridges connecting Cold Island to Air, Water, Earth and Gold islands. For Air Island, Plant and Cold islands have already connected to it, so we need 3 bridges to connect it to Water, Earth and Gold islands. 2 bridges are needed for Water Island, and just one bridge is needed for Earth Island. 5+4+3+2+1 = 15 2-element monsters. The reason why the first line of thinking is wrong is because it treats Earth + Plant and Plant + Earth as 2 different things, while we want them to be the same thing. You might realize that 15 is half of 30. This line of thinking makes sense when you think that every bridge from one island to another can go two different ways, and the "wrong" way of thinking about this makes sense. For MSM, you could do (5*4)/2 = 20/2 = 10, which is the number of two-element monsters in MSM. Notice that 15 and 10 are both triangular numbers. 15 is the fifth triangular number, and 10 is the fourth. The fourth triangular number is (5*4)/2, which can also be written as 5*(4/2). If 4 is replaced with N, then it's (n+1)*(n/2). This relates My Singing Monsters to Gauss's theroem. Anyway, three element monsters. I mentioned that the 6*5*4 way, which would be 120, wouldn't work. Instead of working with tetrahedral numbers, you could just say that each three-element combination (for example: Earth + Cold + Fire) can be written in 6 different ways, because 3! = 6. So: ECF EFC FEC FCE CEF CFE So you could divide 120 by 6, since every trio of three elements appears 6 times in the "all permutations" method. So that would equal 20. There you go. There are 20 possible 3-element monsters. As for 4-element monsters, you could do (6*5*4*3)/4!, but there's an easier way. Each four-element monster has a unique set of two elements missing, and those two elements can be used to make a two-element monster. Because we already figured out that there are 15 2-element monsters, there are also 15 4-element monsters. 5-element monsters are calculated just the same. Each of them has 1 unique element missing. Because there are 6 elements, there are also 6 5-element monsters. And there's only one 6-element monster because there are only 6 elements. 6 + 15 + 20 + 15 + 6 + 1 makes a grand total of 63 monsters that BBB can add in MSM: DoF. << Previous MSMath: Well, there is none, because it's the first post. >> Next MSMath: Beds in DoF (Coming Soon (I hope)) Category:Blog posts